/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![36, 6, -13, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {4,2,1/3*w^3 + 2/3*w^2 - 10/3*w - 7}>;

primesArray := [
[4, 2, 1/3*w^3 - 4/3*w^2 - 4/3*w + 6],
[4, 2, -1/3*w^3 - 2/3*w^2 + 10/3*w + 7],
[9, 3, -1/2*w^3 + 3/2*w^2 + 7/2*w - 9],
[9, 3, 1/3*w^3 + 2/3*w^2 - 7/3*w - 5],
[11, 11, -1/6*w^3 + 1/6*w^2 + 1/6*w],
[19, 19, w + 1],
[19, 19, 1/6*w^3 - 1/6*w^2 - 13/6*w + 2],
[25, 5, -1/3*w^3 + 1/3*w^2 + 7/3*w - 1],
[29, 29, -1/6*w^3 + 1/6*w^2 + 13/6*w],
[29, 29, w - 1],
[31, 31, 1/3*w^3 - 1/3*w^2 - 10/3*w + 3],
[31, 31, 1/6*w^3 - 1/6*w^2 - 1/6*w + 2],
[41, 41, -1/2*w^3 - 1/2*w^2 + 9/2*w + 5],
[41, 41, -1/2*w^3 + 3/2*w^2 + 5/2*w - 8],
[59, 59, 1/3*w^3 - 1/3*w^2 - 10/3*w - 1],
[59, 59, -1/3*w^3 + 4/3*w^2 + 7/3*w - 7],
[59, 59, 1/2*w^3 - 1/2*w^2 - 5/2*w + 2],
[79, 79, w^2 - 11],
[79, 79, 1/6*w^3 - 7/6*w^2 - 7/6*w + 3],
[89, 89, -1/6*w^3 + 1/6*w^2 + 19/6*w - 5],
[89, 89, 1/6*w^3 - 1/6*w^2 - 19/6*w - 3],
[109, 109, -1/6*w^3 + 7/6*w^2 + 1/6*w - 9],
[109, 109, -7/6*w^3 + 19/6*w^2 + 49/6*w - 20],
[109, 109, -5/6*w^3 + 17/6*w^2 + 23/6*w - 12],
[109, 109, 1/6*w^3 + 5/6*w^2 - 13/6*w - 4],
[121, 11, 1/6*w^3 - 1/6*w^2 - 7/6*w - 3],
[139, 139, 1/3*w^3 - 4/3*w^2 + 5/3*w - 1],
[139, 139, -5/3*w^3 - 7/3*w^2 + 50/3*w + 29],
[169, 13, w^3 + w^2 - 10*w - 13],
[169, 13, -5/6*w^3 + 17/6*w^2 + 17/6*w - 12],
[179, 179, -1/3*w^3 + 4/3*w^2 + 1/3*w - 7],
[179, 179, -2/3*w^3 + 2/3*w^2 + 17/3*w - 5],
[181, 181, -w^3 + 3*w^2 + 4*w - 13],
[181, 181, -1/6*w^3 + 7/6*w^2 + 1/6*w - 11],
[181, 181, 7/6*w^3 + 11/6*w^2 - 79/6*w - 25],
[181, 181, 1/6*w^3 + 5/6*w^2 - 13/6*w - 2],
[191, 191, -5/6*w^3 - 1/6*w^2 + 53/6*w + 7],
[191, 191, 1/6*w^3 + 11/6*w^2 - 19/6*w - 16],
[211, 211, -1/6*w^3 + 1/6*w^2 - 5/6*w - 2],
[211, 211, -1/2*w^3 + 1/2*w^2 + 9/2*w + 2],
[211, 211, 1/3*w^3 - 1/3*w^2 - 4/3*w - 3],
[211, 211, -1/2*w^3 + 1/2*w^2 + 11/2*w - 4],
[229, 229, 5/6*w^3 + 1/6*w^2 - 53/6*w - 9],
[229, 229, 1/2*w^3 - 3/2*w^2 - 1/2*w + 2],
[239, 239, 5/6*w^3 + 7/6*w^2 - 59/6*w - 19],
[239, 239, 1/2*w^3 - 5/2*w^2 + 1/2*w + 5],
[241, 241, 1/6*w^3 - 1/6*w^2 - 13/6*w - 4],
[241, 241, w - 5],
[269, 269, -1/6*w^3 + 7/6*w^2 + 1/6*w - 2],
[269, 269, 1/3*w^3 - 7/3*w^2 - 1/3*w + 7],
[271, 271, 5/3*w^3 + 7/3*w^2 - 47/3*w - 25],
[271, 271, -5/3*w^3 + 17/3*w^2 + 23/3*w - 27],
[281, 281, 1/6*w^3 + 5/6*w^2 - 1/6*w - 7],
[281, 281, -1/2*w^3 + 3/2*w^2 + 9/2*w - 8],
[311, 311, 11/6*w^3 + 7/6*w^2 - 113/6*w - 23],
[311, 311, -4/3*w^3 + 13/3*w^2 + 10/3*w - 13],
[311, 311, -2/3*w^3 + 5/3*w^2 + 20/3*w - 17],
[311, 311, 1/6*w^3 + 5/6*w^2 + 5/6*w + 1],
[349, 349, -1/2*w^3 + 1/2*w^2 + 11/2*w - 1],
[349, 349, -5/6*w^3 + 17/6*w^2 + 29/6*w - 14],
[349, 349, 1/6*w^3 - 1/6*w^2 + 5/6*w - 1],
[349, 349, -2/3*w^3 - 4/3*w^2 + 17/3*w + 13],
[359, 359, -1/6*w^3 + 13/6*w^2 + 7/6*w - 18],
[359, 359, -1/6*w^3 - 11/6*w^2 + 13/6*w + 18],
[361, 19, 2/3*w^3 - 2/3*w^2 - 14/3*w + 1],
[379, 379, 1/2*w^3 + 1/2*w^2 - 7/2*w - 7],
[379, 379, -5/6*w^3 + 11/6*w^2 + 23/6*w - 7],
[379, 379, w^3 - 9*w - 5],
[379, 379, -2/3*w^3 + 5/3*w^2 + 14/3*w - 7],
[401, 401, -1/3*w^3 + 7/3*w^2 + 4/3*w - 17],
[401, 401, -4/3*w^3 - 8/3*w^2 + 43/3*w + 31],
[419, 419, 1/6*w^3 - 1/6*w^2 + 5/6*w],
[419, 419, 1/2*w^3 - 1/2*w^2 - 11/2*w + 2],
[421, 421, -5/6*w^3 - 7/6*w^2 + 47/6*w + 11],
[421, 421, 1/3*w^3 - 1/3*w^2 - 19/3*w - 7],
[431, 431, 5/6*w^3 + 1/6*w^2 - 41/6*w - 6],
[431, 431, 2/3*w^3 - 8/3*w^2 - 8/3*w + 15],
[431, 431, 5/6*w^3 - 11/6*w^2 - 29/6*w + 7],
[431, 431, 1/2*w^3 - 5/2*w^2 + 5/2*w + 2],
[439, 439, 5/6*w^3 + 1/6*w^2 - 41/6*w - 4],
[439, 439, 1/2*w^3 - 1/2*w^2 - 9/2*w + 8],
[449, 449, 5/6*w^3 - 17/6*w^2 - 29/6*w + 18],
[449, 449, 11/6*w^3 - 41/6*w^2 - 41/6*w + 29],
[449, 449, -7/6*w^3 + 31/6*w^2 + 7/6*w - 16],
[449, 449, 5/3*w^3 + 7/3*w^2 - 56/3*w - 33],
[461, 461, -1/6*w^3 + 1/6*w^2 + 19/6*w - 3],
[461, 461, 1/6*w^3 - 1/6*w^2 - 19/6*w - 1],
[479, 479, -2*w - 1],
[479, 479, 1/3*w^3 - 1/3*w^2 - 13/3*w + 3],
[491, 491, 2/3*w^3 + 1/3*w^2 - 14/3*w - 7],
[491, 491, 5/6*w^3 - 11/6*w^2 - 35/6*w + 7],
[509, 509, 5/6*w^3 - 11/6*w^2 - 29/6*w + 8],
[509, 509, 5/6*w^3 + 1/6*w^2 - 41/6*w - 5],
[521, 521, -1/6*w^3 + 7/6*w^2 - 17/6*w + 1],
[521, 521, -1/6*w^3 + 7/6*w^2 - 11/6*w - 3],
[521, 521, -1/6*w^3 + 7/6*w^2 + 19/6*w - 5],
[521, 521, 2/3*w^3 + 1/3*w^2 - 26/3*w - 11],
[541, 541, 1/6*w^3 + 5/6*w^2 - 31/6*w - 17],
[541, 541, 5/6*w^3 - 17/6*w^2 - 47/6*w + 23],
[541, 541, 3*w^3 + 3*w^2 - 32*w - 47],
[541, 541, 2/3*w^3 + 4/3*w^2 - 14/3*w - 11],
[571, 571, -1/3*w^3 + 1/3*w^2 + 13/3*w - 1],
[571, 571, 2*w - 1],
[571, 571, -1/3*w^3 + 4/3*w^2 + 7/3*w - 1],
[571, 571, 1/6*w^3 + 5/6*w^2 - 7/6*w - 13],
[599, 599, 1/2*w^3 + 1/2*w^2 - 9/2*w - 2],
[599, 599, -1/6*w^3 + 1/6*w^2 + 19/6*w - 2],
[599, 599, 1/6*w^3 - 1/6*w^2 - 19/6*w],
[599, 599, 1/2*w^3 - 3/2*w^2 - 5/2*w + 11],
[601, 601, 1/6*w^3 + 11/6*w^2 - 7/6*w - 15],
[601, 601, -1/3*w^3 + 7/3*w^2 - 5/3*w - 5],
[601, 601, 2/3*w^3 + 4/3*w^2 - 26/3*w - 19],
[601, 601, 1/2*w^3 - 5/2*w^2 - 7/2*w + 13],
[659, 659, w^2 - 13],
[659, 659, -5/6*w^3 + 11/6*w^2 + 29/6*w - 12],
[659, 659, -1/6*w^3 + 7/6*w^2 + 7/6*w - 1],
[659, 659, 5/6*w^3 + 1/6*w^2 - 41/6*w - 1],
[691, 691, 1/2*w^3 - 1/2*w^2 - 11/2*w + 10],
[691, 691, -1/6*w^3 + 1/6*w^2 - 5/6*w - 8],
[701, 701, -2*w^2 + w + 11],
[701, 701, -1/6*w^3 + 13/6*w^2 + 7/6*w - 9],
[701, 701, -1/6*w^3 + 13/6*w^2 + 1/6*w - 16],
[701, 701, -w^3 + w^2 + 9*w - 1],
[709, 709, -2/3*w^3 + 11/3*w^2 + 14/3*w - 21],
[709, 709, 1/3*w^3 + 5/3*w^2 + 2/3*w - 5],
[709, 709, 7/6*w^3 - 25/6*w^2 - 43/6*w + 25],
[709, 709, 2/3*w^3 - 2/3*w^2 - 23/3*w - 1],
[719, 719, 2*w^2 - 2*w - 19],
[719, 719, -2*w^2 + 2*w + 7],
[739, 739, 5/6*w^3 - 5/6*w^2 - 29/6*w + 4],
[739, 739, -5/6*w^3 + 11/6*w^2 + 53/6*w - 17],
[739, 739, -1/6*w^3 - 5/6*w^2 - 11/6*w],
[739, 739, -w^3 + w^2 + 8*w - 5],
[751, 751, -1/6*w^3 + 1/6*w^2 + 25/6*w + 8],
[751, 751, -1/3*w^3 + 1/3*w^2 + 16/3*w - 11],
[761, 761, -2/3*w^3 + 5/3*w^2 + 8/3*w - 9],
[761, 761, 5/6*w^3 + 1/6*w^2 - 47/6*w - 3],
[769, 769, 1/3*w^3 - 7/3*w^2 - 7/3*w + 11],
[769, 769, -4/3*w^3 - 8/3*w^2 + 46/3*w + 33],
[769, 769, -1/2*w^3 + 5/2*w^2 + 5/2*w - 20],
[769, 769, 2*w^2 - 17],
[809, 809, 1/6*w^3 + 11/6*w^2 - 31/6*w - 15],
[809, 809, -5/6*w^3 - 1/6*w^2 + 35/6*w + 3],
[809, 809, 1/6*w^3 + 11/6*w^2 - 31/6*w - 9],
[809, 809, w^3 - 2*w^2 - 7*w + 11],
[811, 811, -1/6*w^3 - 5/6*w^2 + 25/6*w + 7],
[811, 811, 1/6*w^3 + 5/6*w^2 - 25/6*w - 15],
[811, 811, -1/6*w^3 - 5/6*w^2 + 25/6*w - 4],
[811, 811, 1/6*w^3 + 5/6*w^2 - 25/6*w - 4],
[821, 821, -1/2*w^3 - 5/2*w^2 + 13/2*w + 16],
[821, 821, 1/6*w^3 - 13/6*w^2 - 7/6*w + 21],
[821, 821, 3/2*w^3 + 1/2*w^2 - 35/2*w - 20],
[821, 821, -1/6*w^3 + 13/6*w^2 + 7/6*w - 7],
[829, 829, -1/6*w^3 + 19/6*w^2 - 11/6*w - 23],
[829, 829, 5/6*w^3 - 17/6*w^2 - 5/6*w + 8],
[829, 829, -4/3*w^3 - 2/3*w^2 + 43/3*w + 15],
[829, 829, -1/6*w^3 - 17/6*w^2 + 25/6*w + 16],
[839, 839, 1/3*w^3 - 7/3*w^2 + 2/3*w + 5],
[839, 839, 1/2*w^3 + 3/2*w^2 - 13/2*w - 20],
[841, 29, -5/6*w^3 + 5/6*w^2 + 35/6*w - 4],
[859, 859, 1/6*w^3 + 5/6*w^2 - 37/6*w + 7],
[859, 859, 13/6*w^3 + 17/6*w^2 - 127/6*w - 33],
[881, 881, -2/3*w^3 + 2/3*w^2 + 20/3*w + 5],
[881, 881, 17/6*w^3 + 19/6*w^2 - 167/6*w - 42],
[911, 911, -1/6*w^3 + 13/6*w^2 - 11/6*w - 6],
[911, 911, 3/2*w^3 + 1/2*w^2 - 33/2*w - 17],
[911, 911, 5/6*w^3 - 17/6*w^2 + 1/6*w + 5],
[911, 911, 7/6*w^3 - 1/6*w^2 - 61/6*w - 1],
[929, 929, -7/6*w^3 - 11/6*w^2 + 67/6*w + 22],
[929, 929, -1/6*w^3 + 1/6*w^2 - 11/6*w + 6],
[941, 941, 11/6*w^3 + 19/6*w^2 - 107/6*w - 34],
[941, 941, 4/3*w^3 - 10/3*w^2 - 25/3*w + 19],
[941, 941, -11/6*w^3 + 41/6*w^2 + 47/6*w - 31],
[941, 941, 1/2*w^3 + 1/2*w^2 - 13/2*w - 13],
[961, 31, -1/3*w^3 + 1/3*w^2 + 7/3*w + 5],
[991, 991, -1/6*w^3 - 5/6*w^2 + 25/6*w + 6],
[991, 991, 1/6*w^3 + 5/6*w^2 - 25/6*w - 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 - 3*x - 2;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1, 3, -e - 2, 2, -e + 7, -2*e + 2, e - 7, -e + 6, -2*e + 1, -2*e + 4, 4*e - 4, -2*e + 3, -e + 8, -3*e + 13, 2*e, -5*e + 3, 5*e - 3, -2*e, 5*e - 4, 9, 3*e - 10, -2*e + 3, 5*e, 5*e, 7*e - 10, -6, -3*e + 17, -e, 11*e - 16, -4*e + 10, -e - 13, -7*e + 16, -2*e, -6*e + 21, 3*e - 13, e + 11, 3*e - 17, -3*e + 9, 4*e - 4, -9*e + 7, 5*e + 11, 17, 7*e - 24, -18, -4*e, -4*e + 3, 7*e - 18, -5*e + 8, 7*e - 8, -e - 3, 10*e - 24, -3*e + 20, -e - 8, -4*e - 4, 7*e - 25, -e - 17, -4*e + 6, e - 3, -3*e + 30, -6*e, 2*e - 21, -3*e - 21, 12*e - 22, 14*e - 27, -4*e - 2, 10*e - 18, -13*e + 19, -14*e + 24, -2*e + 8, 18, 2*e - 34, 8*e - 4, 3*e + 16, 6*e - 7, -3*e - 25, e + 13, 4*e + 10, e - 25, 4*e - 8, -7*e + 13, -3*e + 14, 4*e - 27, 17*e - 27, -2*e - 8, -9*e + 14, -5*e - 4, 12*e - 6, -6*e - 20, -14*e + 24, -5*e - 7, -20*e + 29, 3*e - 8, e + 2, 6, -19*e + 25, -9*e + 28, 3*e - 20, e + 14, 5*e + 28, e - 24, e - 19, 4*e - 4, -2*e + 8, -4*e - 2, -10*e - 2, 13*e - 15, 2*e + 6, -2*e, -5*e + 12, 6*e - 16, -16*e + 26, e + 4, -17*e + 37, -2*e - 4, -4*e + 26, 9*e + 13, 2*e - 2, -3*e + 49, -13*e + 6, -8*e + 38, -11*e + 16, -10*e + 28, 13*e - 24, -8*e + 3, -3*e + 28, -4*e + 43, -6*e - 6, 16*e - 10, 3*e - 1, 7*e + 27, -e - 13, -4*e + 2, 17*e - 37, 2*e - 36, 7*e - 33, -16*e + 42, -7*e + 20, 12*e - 7, 10*e - 17, -21*e + 26, 10, -12*e + 3, 4*e - 46, 2*e - 3, 17*e - 41, 10*e + 14, 2*e + 12, -12*e + 42, -2*e + 8, 6*e + 12, -12*e + 34, 2*e - 8, -2*e + 24, 17*e - 24, 9*e + 12, -12*e + 50, -19*e + 21, 6*e - 6, -12*e + 2, 2*e - 30, -6*e + 6, -14*e + 5, -9*e + 30, 16*e - 26, -18*e + 16, 2*e + 40, 2*e - 20, -2*e + 4, 20*e - 38, -21*e + 42, -11*e + 10, -17*e + 24, 24*e - 43, -e - 29, 8*e - 36, -10*e + 26];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4,2,1/3*w^3 + 2/3*w^2 - 10/3*w - 7}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;